## Abstract

We continue our study in [16] of the Gamma limit of the Abelian ChernSimonsHiggs energy G_{csh} := 1/2 ∫_{U} |∇_{Aε} u_{ε}|^{2} + μ_{ε}^{2}/4 |curl A_{ε} - h _{ex}|^{2}/|u_{ε}|^{2} + 1/ε^{2} |u_{ε}|^{2} ( 1 - |u _{ε}|^{2})^{2} dx on a bounded, simply connected, two-dimensional domain where ε → 0 and μ(ε) → μ ∈ [0, +∞]. Under the critical scaling, G(csh) ≈ |log ε|^{2} , we establish the Gamma limit when μ ∈ (0,+∞], and as a consequence, we are able to compute the first critical field H"1 = H"1(U,μ) for the nucleation of a vortex. Finally, we show failure of Gamma convergence when μ(μ) → 0 (this includes the self-dual case). The method entails estimating in certain weak topologies the Jacobian J(u(ε)) = det(∇ u(ε)) in terms of the ChernSimonsHiggs energy E(csh).

Original language | English (US) |
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Pages (from-to) | 1-16 |

Number of pages | 16 |

Journal | Communications in Contemporary Mathematics |

Volume | 10 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2008 |

### Bibliographical note

Funding Information:D. Spirn was supported in part by NSF grant DMS–0510121.

## Keywords

- Chern-Simons-Higgs theory
- Vortices